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A remark on central sequence algebras of the tensor product of $ \mathrm{II}_{1}$ factors

Authors: Wenming Wu and Wei Yuan
Journal: Proc. Amer. Math. Soc. 142 (2014), 2829-2835
MSC (2010): Primary 46L10
Published electronically: May 12, 2014
MathSciNet review: 3209336
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Abstract: Let $ \mathcal {M}$ and $ \mathcal {N}$ be two type $ \mathrm {II}_{1}$ factors with separable predual and $ \omega $ a free ultrafilter on $ \mathbb{N}$. If the central sequence algebra $ \mathcal {N}_{\omega }$ is abelian and there is a non-atomic abelian subalgebra $ \mathcal {A}$ in $ \mathcal {M}$ such that any central sequence of $ \mathcal {M}\overline {\otimes }\mathcal {N}$ is contained in the ultrapower $ (\mathcal {A}\overline {\otimes }\mathcal {N})^{\omega }$, then $ (\mathcal {M}\overline {\otimes }\mathcal {N})_{\omega }$ is abelian. It is also shown that there is an action $ \alpha $ of the free group $ F_2$ on the group von Neumann algebra $ \mathcal {L}_{\mathbb{Z}}$ such that the central sequence algebra of $ \mathcal {M}=\mathcal {L}_{\mathbb{Z}}\rtimes _{\alpha } F_2$ is abelian and non-trivial and any central sequence in $ \mathcal {M}\overline {\otimes }\mathcal {N}$ is in the ultrapower $ (\mathcal {L}_{\mathbb{Z}}\overline {\otimes }\mathcal {N})^{\omega }$.

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  • [1] A. Connes, Classification of injective factors. Cases $ II_{1},$ $ II_{\infty },$ $ III_{\lambda },$ $ \lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73-115. MR 0454659 (56 #12908)
  • [2] J. Dixmier, Quelques propriétés des suites centrales dans les facteurs de type $ {\rm II}_{1}$, Invent. Math. 7 (1969), 215-225 (French). MR 0248534 (40 #1786)
  • [3] Junsheng Fang, Liming Ge, and Weihua Li, Central sequence algebras of von Neumann algebras, Taiwanese J. Math. 10 (2006), no. 1, 187-200. MR 2186173 (2006k:46094)
  • [4] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, Pure and Applied Mathematics, vol. 100, Academic Press Inc., Orlando, FL, 1986. MR 859186 (88d:46106)
  • [5] Dusa McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443-461. MR 0281018 (43 #6737)
  • [6] Narutaka Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111-117. MR 2079600 (2005e:46115),
  • [7] Narutaka Ozawa, A Kurosh-type theorem for type $ \rm II_1$ factors, Int. Math. Res. Not. 2006, Art. ID 97560, 18 pp.. MR 2211141 (2006m:46078),
  • [8] Sorin Popa, Orthogonal pairs of $ \ast $-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253-268. MR 703810 (84h:46077)
  • [9] S. Sakai, The theory of W*-algebras, Lecture notes, Yale University, 1962.
  • [10] M. Takesaki, Theory of operator algebras. I, reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences, vol. 124, Operator Algebras and Non-commutative Geometry, 5, Springer-Verlag, Berlin, 2002. MR 1873025 (2002m:46083)
  • [11] Elias M. Stein and Rami Shakarchi, Real analysis, Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, III, Princeton University Press, Princeton, NJ, 2005. MR 2129625 (2005k:28024)

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Additional Information

Wenming Wu
Affiliation: College of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, People’s Republic of China

Wei Yuan
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, 100084, People’s Republic of China

Received by editor(s): November 18, 2011
Received by editor(s) in revised form: September 10, 2012
Published electronically: May 12, 2014
Additional Notes: This work was partially supported by NSFC (No.11271390, No. 11301511) and Natural Science Foundation Project of CQ CSTC (No. CSTC, 2010BB9318).
Communicated by: Marius Junge
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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